Solve -3x 7y=5x^2y

Question

Solve the equation

Solve for x

Solve for y

x = 0 x = - \frac{21}{5}

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Testing for symmetry

Testing for symmetry about the origin

Testing for symmetry about the x-axis

Testing for symmetry about the y-axis

Not symmetry with respect to the origin

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Rewrite the equation

r = 0 r = - \frac{21 sec ( θ )}{5}

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Find the first derivative

Find the derivative with respect to x

Find the derivative with respect to y

\frac{d y}{d x} = - \frac{10 x y + 21 y}{21 x + 5 x ^{2}}

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Find the second derivative

Find the second derivative with respect to x

Find the second derivative with respect to y

\frac{d ^{2} y}{d x ^{2}} = \frac{882 y + 630 y x + 150 y x ^{2}}{441 x ^{2} + 210 x ^{3} + 25 x ^{4}}

Calculate

- 3 x 7 y = 5 x^{2} y

Simplify the expression

- 21 x y = 5 x^{2} y

Take the derivative of both sides

\frac{d}{d x} (- 21 x y) = \frac{d}{d x} (5 x^{2} y)

Calculate the derivative

More Steps

- 21 y - 21 x \frac{d y}{d x} = \frac{d}{d x} (5 x^{2} y)

Calculate the derivative

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- 21 y - 21 x \frac{d y}{d x} = 10 x y + 5 x^{2} \frac{d y}{d x}

Move the expression to the left side

- 21 y - 21 x \frac{d y}{d x} - 5 x^{2} \frac{d y}{d x} = 10 x y

Move the expression to the right side

- 21 x \frac{d y}{d x} - 5 x^{2} \frac{d y}{d x} = 10 x y + 21 y

Collect like terms by calculating the sum or difference of their coefficients

(- 21 x - 5 x^{2}) \frac{d y}{d x} = 10 x y + 21 y

Divide both sides

\frac{( - 21 x - 5 x ^{2} ) \frac{d y}{d x}}{- 21 x - 5 x ^{2}} = \frac{10 x y + 21 y}{- 21 x - 5 x ^{2}}

Divide the numbers

\frac{d y}{d x} = \frac{10 x y + 21 y}{- 21 x - 5 x ^{2}}

Use \frac{- a}{b} = \frac{a}{- b} = - \frac{a}{b} to rewrite the fraction

\frac{d y}{d x} = - \frac{10 x y + 21 y}{21 x + 5 x ^{2}}

Take the derivative of both sides

\frac{d}{d x} (\frac{d y}{d x}) = \frac{d}{d x} (- \frac{10 x y + 21 y}{21 x + 5 x ^{2}})

Calculate the derivative

\frac{d ^{2} y}{d x ^{2}} = \frac{d}{d x} (- \frac{10 x y + 21 y}{21 x + 5 x ^{2}})

Use differentiation rules

\frac{d ^{2} y}{d x ^{2}} = - \frac{\frac{d}{d x} ( 10 x y + 21 y ) \times ( 21 x + 5 x ^{2} ) - ( 10 x y + 21 y ) \times \frac{d}{d x} ( 21 x + 5 x ^{2} )}{( 21 x + 5 x ^{2} ) ^{2}}

Calculate the derivative

More Steps

\frac{d ^{2} y}{d x ^{2}} = - \frac{( 10 y + 10 x \frac{d y}{d x} + 21 \frac{d y}{d x} ) ( 21 x + 5 x ^{2} ) - ( 10 x y + 21 y ) \times \frac{d}{d x} ( 21 x + 5 x ^{2} )}{( 21 x + 5 x ^{2} ) ^{2}}

Calculate the derivative

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\frac{d ^{2} y}{d x ^{2}} = - \frac{( 10 y + 10 x \frac{d y}{d x} + 21 \frac{d y}{d x} ) ( 21 x + 5 x ^{2} ) - ( 10 x y + 21 y ) ( 21 + 10 x )}{( 21 x + 5 x ^{2} ) ^{2}}

Calculate

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\frac{d ^{2} y}{d x ^{2}} = - \frac{210 y x + 50 y x ^{2} + 315 x ^{2} \frac{d y}{d x} + 50 x ^{3} \frac{d y}{d x} + 441 x \frac{d y}{d x} - ( 10 x y + 21 y ) ( 21 + 10 x )}{( 21 x + 5 x ^{2} ) ^{2}}

Calculate

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\frac{d ^{2} y}{d x ^{2}} = - \frac{210 y x + 50 y x ^{2} + 315 x ^{2} \frac{d y}{d x} + 50 x ^{3} \frac{d y}{d x} + 441 x \frac{d y}{d x} - ( 420 x y + 441 y + 100 x ^{2} y )}{( 21 x + 5 x ^{2} ) ^{2}}

Calculate

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\frac{d ^{2} y}{d x ^{2}} = - \frac{- 210 y x - 50 y x ^{2} + 315 x ^{2} \frac{d y}{d x} + 50 x ^{3} \frac{d y}{d x} + 441 x \frac{d y}{d x} - 441 y}{( 21 x + 5 x ^{2} ) ^{2}}

Use equation \frac{d y}{d x} = - \frac{10 x y + 21 y}{21 x + 5 x ^{2}} to substitute

\frac{d ^{2} y}{d x ^{2}} = - \frac{- 210 y x - 50 y x ^{2} + 315 x ^{2} ( - \frac{10 x y + 21 y}{21 x + 5 x ^{2}} ) + 50 x ^{3} ( - \frac{10 x y + 21 y}{21 x + 5 x ^{2}} ) + 441 x ( - \frac{10 x y + 21 y}{21 x + 5 x ^{2}} ) - 441 y}{( 21 x + 5 x ^{2} ) ^{2}}

Solution

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Calculate

- \frac{- 210 y x - 50 y x ^{2} + 315 x ^{2} ( - \frac{10 x y + 21 y}{21 x + 5 x ^{2}} ) + 50 x ^{3} ( - \frac{10 x y + 21 y}{21 x + 5 x ^{2}} ) + 441 x ( - \frac{10 x y + 21 y}{21 x + 5 x ^{2}} ) - 441 y}{( 21 x + 5 x ^{2} ) ^{2}}

Multiply

More Steps

- \frac{- 210 y x - 50 y x ^{2} - \frac{315 x ( 10 x y + 21 y )}{21 + 5 x} + 50 x ^{3} ( - \frac{10 x y + 21 y}{21 x + 5 x ^{2}} ) + 441 x ( - \frac{10 x y + 21 y}{21 x + 5 x ^{2}} ) - 441 y}{( 21 x + 5 x ^{2} ) ^{2}}

Multiply

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- \frac{- 210 y x - 50 y x ^{2} - \frac{315 x ( 10 x y + 21 y )}{21 + 5 x} - \frac{50 x ^{2} ( 10 x y + 21 y )}{21 + 5 x} + 441 x ( - \frac{10 x y + 21 y}{21 x + 5 x ^{2}} ) - 441 y}{( 21 x + 5 x ^{2} ) ^{2}}

Multiply

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- \frac{- 210 y x - 50 y x ^{2} - \frac{315 x ( 10 x y + 21 y )}{21 + 5 x} - \frac{50 x ^{2} ( 10 x y + 21 y )}{21 + 5 x} - \frac{441 ( 10 x y + 21 y )}{21 + 5 x} - 441 y}{( 21 x + 5 x ^{2} ) ^{2}}

Subtract the terms

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- \frac{- 882 y - 630 y x - 150 y x ^{2}}{( 21 x + 5 x ^{2} ) ^{2}}

Use \frac{- a}{b} = \frac{a}{- b} = - \frac{a}{b} to rewrite the fraction

- (- \frac{882 y + 630 y x + 150 y x ^{2}}{( 21 x + 5 x ^{2} ) ^{2}})

Calculate

\frac{882 y + 630 y x + 150 y x ^{2}}{( 21 x + 5 x ^{2} ) ^{2}}

Expand the expression

More Steps

\frac{882 y + 630 y x + 150 y x ^{2}}{441 x ^{2} + 210 x ^{3} + 25 x ^{4}}

\frac{d ^{2} y}{d x ^{2}} = \frac{882 y + 630 y x + 150 y x ^{2}}{441 x ^{2} + 210 x ^{3} + 25 x ^{4}}

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